I thought if you make too big of a water change it can mess with the cycling process or an established cycle. Is that not true?

No that is not true. Water changes have no bearing on the cycle - the bacteria lives on the surfaces in the tank and filter, not in the water itself. You can change as much water as you wish, so long as the chemistry isn't too different.

While you are in the process of a fishless cycle, water changes are not as important. The bacteria can grow in ammonia concentrations that are lethal for fish. During a fishless cycle there is a time when a water change may be needed, and that's after the ammonia has been converted to nitrite. If the nitrite concentration is too high, it will cause the cycle process to stall.

During a fish in cycle, the goal should be to keep the ammonia level as low as possible so that the fish is not poisoned, so large frequent water changes are needed.

No that is not true. Water changes have no bearing on the cycle - the bacteria lives on the surfaces in the tank and filter, not in the water itself. You can change as much water as you wish, so long as the chemistry isn't too different.

While you are in the process of a fishless cycle, water changes are not as important. The bacteria can grow in ammonia concentrations that are lethal for fish. During a fishless cycle there is a time when a water change may be needed, and that's after the ammonia has been converted to nitrite. If the nitrite concentration is too high, it will cause the cycle process to stall.

During a fish in cycle, the goal should be to keep the ammonia level as low as possible so that the fish is not poisoned, so large frequent water changes are needed.

Roughly, that would be accurate. If your tap water has ammonia in it (test it!), you may need more. Keep in mind, Prime converts the ammonia to ammonium, so certain test kits (like the API Freshwater Master Test Kit so many of us have) will still show ammonia in the water because the test cannot distinguish between ammonia and ammonium.

Prime uses a sulfur compound (hence the smell) to add a hydrogen ion to ammonia (NH3) converting it to ammonium (NH4). Ammonium only exists at <6.5pH. So above that, as soon as it's converted, the ammonium wants to change back to ammonia. The higher the pH the faster this happens. This is a simplified explanation.

As I remember it, Seachem says it stays converted from 24 to 48 hours. This is easier than trying to explain subtle chemistry they hardly understand. Their explanations are rather poorly framed and explicated in my opinion.